# Thread: Sum and products of series

1. ## Sum and products of series

Hello,

I was wondering if anyone could help me show the following:

$\displaystyle \displaystyle\sum_{i=1}^n\frac{(t+n-1)!(i-1)!}{(t+i-1)!}=\frac{(n+t-1)!-n!}{(t-1)!(t-1)}$,

where $\displaystyle t\geqslant2$.

It's an important part of a larger problem I'm working on and I'm a little stuck.

2. Originally Posted by hairymclairy
Hello,

I was wondering if anyone could help me show the following:

$\displaystyle \displaystyle\sum_{i=1}^n\frac{(t+n-1)!(i-1)!}{(t+i-1)!}=\frac{(n+t-1)!-n!}{(t-1)!(t-1)}$,

where $\displaystyle t\geqslant2$.

It's an important part of a larger problem I'm working on and I'm a little stuck.

Induction on n.

If we checking this formula when n=1:

{(t+1-1)!(1-1)!}/{(t+1-1)!}=t!/t!=1

And:

{(1+t-1)!-1!}/{(t-1)!(t-1)}={t!-1}/{t!-1}=1

Try to continue...

3. I know it's provable by induction, however I need to derive the right hand side from the left, not just prove it if you see what I mean. Maybe the problem should be rephrased to "find a closed expression (i.e. an expression not involving the summation and product symbols) for the left hand side of that equation".

4. The question asked me to derive various formulas, but when I spoke to my supervisor he said that it's apparently fine to just find a formula and then prove by induction, so thanks!